量子化学3

Cu = ABu, C = AB
∧ ∧ ∧ ∧ ∧ ∧
•The identity operator does nothing (or multiplies by 1)
1u = u
∧
•The associative law holds for operators
A(BC) = (AB) C
∧ ∧ ∧ ∧ ∧ ∧
∧
Operator Operation Multiply by r
P = −ih i ∂ ∂r i
r
∧
Pi
∧
T
∧
∂2 ∂2 h2 ∂2 − + + 2m ∂x2 ∂y2 ∂z2
V (r)
∧
Multiply by V(r)
∂2 ∂2 ∧ h2 ∂2 + V (x, y, z) − + + 2m ∂x2 ∂y2 ∂z2 ∂ ∂ − ih y ∂z − z ∂y ∂ ∂ − ih z −x ∂x ∂z ∂ ∂ − ih x −y ∂x ∂y
DefiniteLapacian then
def ∂2 ∂2 ∂2 2 + 2 + 2 ∇ ∂x ∂y ∂z
2
h2 2 ∧ − 2m∇ + Vψ = Eψ
Definite Hamiltonian
Hψ = Eψ
∧
def h2 2 ∧ H − 2m∇ + V
∧ ∧ ∧ ∧
•The sum and difference of two operators
Cu = Au + Bu, = A+ B C Cu = Au − Bu, = A− B C
∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧
•The product of two operators is defined by
Postulate 2 In any measurement of the observable ∧ associated with operator A , the only values that will ever be observed are the eigenvalues a, which satisfy the eigenvalue equation
A = ∫ψ *A dτ = ∫ (ψ A) * dτ ψ ψ
∧ ∧
Operators A which satisfy this condition are called Hermitian.
∧
Further Reading and Homework Rudiments of Quantum Theory www.chembio.uoguelph.ca\educmat\chm386\rudiment\rudiment.htm A Brief Review of elementary Quantum Chemistry /notes/quantrev/quantrev.html 1. Identify which of the following functions of the operator d/dx:(a) eikx,,(b)cos x,(c)k,(d)kx,(e)e-αx. Gave the corresponding eigenvalue where appropriate. 2. Determine which of the following functions are aigenfunctions of the inversion operator i(which has the effect of making the replacement x to -x):(a)x3-kx(b)coskx,(c) x2+3x-1. State the eigenvalue of i when relevent.
Aψ = aψ
Postulate 3. If a system is in a state described by a wave function ψ , then the average value of the observable ∧ corresponding to A is given by
which is called the commutator of A and B Note that the . ∧ ∧ order matters, If Aand B happen to commute, then
[A, B] = 0
∧ ∧
∧
∧
•The n-th power of an operator A is defined as n successive applications of the operator, e.g.
∧
Au = λu
Matrix description of an eigenvalue equation
∧
2.1.4 Operator Expression of the Time-Independent Schrödinger Equation
∧ h2 ∂2 ∂2 ∂2 2 + 2 + 2 ψ (x, y, z) + Vψ (x, y, z) = Eψ (x, y, z) − 2m ∂x ∂y ∂z
2 Mathematical Background and Postulates of Quantum Mechanics
2.1 Operators •Operator An operator is a symbol that tells you to do something with whatever follows the symbol. e.g. +, −, ×, ÷, ln, sin, d/dx … … An operator is a rule that transforms a given function or vector into another function or vector.
2.3 Hermitian Operators As mentioned previously, the expectation value of an ∧ operator A is given by
A = ∫ψ *A dτ ψ
∧
and all physical observables are represented by such expectation values. Obviously, the value of a physical observable such as energy or density must be real, so we require <A> to be real. This means that we must have <A> = <A>*, or
A( f + g) = A f + A g A(cf ) = c A f
∧ ∧ ∧ ∧ ∧
where c is a constant and f and g are functions. As an example, consider the operators d/dx and ()2. We can see that d/dx is a linear operator because
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∧ n
A u = AAu
∧
∧ 2
∧ ∧
•The exponential of an operator eA is defined via the power series
A A e = 1+ A+ + +... 2! 3 !
A
∧
∧
∧
∧ 2
∧ 3
2.1.2 Linear Operators Almost all operators encountered in quantum mechanics are linear operators. A linear operator is an operator which satisfies the following two conditions:
∧
then
2.2 Postulates of Quantum Mechanics •Postulate 1 The state of a quantum mechanical system is completely specified by a function ψ( r, t ) that depends on the coordinates of the particle(s) and on time. This function, called the wave function or state function, has the important property that ψ*( r, t ) ψ( r, t )dτ is the probability that the particle lies in the volume element dτ located at r at time t. The wavefunction must be single-valued, continuous, and finite.
Fu(x) = v(x)
e.g.
∧
∧
u(x) = x ,
2
d F= , dx
∧
d 2 Fu(x) = x = 2x = v(x) dx
2.1.1 Basic Properties of Operators •Two operators are equal if